It is well known that primes of form $4k+3$, call them $3=q_1 < q_2 < \dots$ satisfy $q_{n+1}/q_n\rightarrow 1$ (and even $q_n=\frac{n}{2\log n}(1+o(1))$). I would be glad to see results of Dusart-type (Strong Bertrand postulate) with concrete bounds for $N$ such that $q_{n+1} < 1.1 q_n$ provided $q_n > N$ or something like this (I am not sure that I need a multiple exactly $1.1$)
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asked Feb 13, 2011 at 23:24
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$\begingroup$ I would suggest contacting Pierre Dusart directly. Whether he has any results in that direction or not, I'm sure he'd know what the state of the art is. $\endgroup$– CharlesCommented Feb 18, 2011 at 18:54
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Have a look at (Satz 9 and the preceeding calculations):
1935-10 P. Erdõs: Über die Primzahlen gewisser arithmetischer Reihen (in German), Math. Z. 39 (1935), 473--491; Zentralblatt 10,293.
Available at: http://www.renyi.hu/~p_erdos/Erdos.html
answered Feb 14, 2011 at 0:38
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$\begingroup$ Thank you! Though this seem to work for intervals $[n,2n]$ and I do not see how to change arguments for $[n,cn]$ with $c$ sufficiently close to 1. $\endgroup$ Commented Feb 14, 2011 at 8:50
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$\begingroup$ It is not known how to make these kinds of elementary arguments work with $c$ arbitrarily close to $1$. I don't know how to get it better than $c=2$ but it might be possible. $\endgroup$ Commented Feb 14, 2011 at 8:59
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$\begingroup$ I would be happy with non-elementary methods (with $L$-functions probably) aswell, if they give results of such type. $\endgroup$ Commented Feb 14, 2011 at 16:36